Gamma Positivity of the Excedance-Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups

Abstract

The Eulerian polynomial \( \mathrm {AExc}_n(t)\) enumerating excedances in the symmetric group \(\mathfrak {S}_n\) is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also gamma positive for all n. We consider \( \mathrm {AExc}_n^+(t)\) and \( \mathrm {AExc}_n^-(t)\), the polynomials which enumerate excedance in the alternating group \(\mathcal {A}_n\) and in \(\mathfrak {S}_n - \mathcal {A}_n\), respectively. We show that \( \mathrm {AExc}_n^+(t)\) is gamma positive iff \(n \ge 5\) is odd. When \(n \ge 4\) is even, \( \mathrm {AExc}_n^+(t)\) is not even palindromic, but we show that it is the sum of two gamma positive summands. An identical statement is true about \( \mathrm {AExc}_n^-(t)\). We extend similar results to the excedance based Eulerian polynomial when enumeration is done over the positive elements in both type B and type D Coxeter groups. Gamma positivity results are known when excedance is enumerated over derangements in \(\mathfrak {S}_n\). We extend some of these to the case when enumeration is done over even and odd derangements in \(\mathfrak {S}_n\).

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Acknowledgements

The first author acknowledges support from a CSIR-SPM Fellowship. The second author acknowledges support from project Grant P07 IR052, given by IRCC, IIT Bombay and from project SERB/F/252/2019-2020 given by the Science and Engineering Research Board (SERB), India.

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Correspondence to Sivaramakrishnan Sivasubramanian.

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Dey, H.K., Sivasubramanian, S. Gamma Positivity of the Excedance-Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups. Ann. Comb. (2020). https://doi.org/10.1007/s00026-020-00511-6

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Mathematics Subject Classification

  • 05A05
  • 05A19
  • 05E15

Keywords

  • Gamma positivity
  • Eulerian polynomial
  • Classical Weyl Groups